I have to say that as described you are talking about a mathematical fact. It not something that is amenable to experimental proof or disproof, it simply is. The experiment asks if the signal does or does not superimpose in linear, non-interfering additive way. You are not testing Fourier decomposition (which is math), but the superposition of your signals (which is a physical property assumed in the mathematics of Fourier decomposition).
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dmckee♦Feb 16 '13 at 15:26

@dmckee Are you saying mathematical "facts" cannot be showed to be true experimentally?
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user13107Feb 17 '13 at 8:13

I am saying that mathematics can be true independently of physics. What you test with the mechanisms suggested below is "Do the physics of the signal include the mathematical prerequisites for Fourier composition?"
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dmckee♦Feb 17 '13 at 16:00

3 Answers
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Idea #2: Build an electronic circuit that separates a periodic input signal (e.g. a square wave) into its component frequencies (using an array of band-pass filters) and then adds these signals back together to get an approximation of the original signal.

Thanks. That seems plausible. I am accepting this now but I hope people will still respond with newer ways f showing this. Most of the attempts so far involve electronics. Perhaps someone can come up with an experiment that doesn't involve building circuits.
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user13107Feb 17 '13 at 8:17

Idea #1: Build a physical representation of the phasor diagram. Here, each frequency component is represented as a rotating vector, and all of these vectors are stacked end-to-end. In a physical representation, each vector could be a metal bar, and they could be geared to rotate at the correct rates.

A friend of mine did a very similar experiment for his college project. The idea was to test the frequency response of a Hi-Fi amplifier by using a delta function. A true delta function has equal amplitudes of all frequencies, so if you take the amplifier output and Fourier transform it to get the spectrum this immediately gives you the frequency response.

Of course you can't generate a delta function, so what he did was to use a rectangular wave with as low a mark space ratio as he could get. Then by Fourier transforming both the input and output and taking the ratio of the spectra you can extract the frequency response.

As I recall, the experiment worked OK but it was hard to get the input pulse narrow enough to get a really wide frequency range. After all, the range of hearing extends for three orders of magnitude.